Optimal. Leaf size=55 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {k^2-2} \sqrt {k^2 x^3+\left (-k^2-1\right ) x^2+x}}{k^2 x-1}\right )}{\sqrt {k^2-2}} \]
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Rubi [C] time = 5.07, antiderivative size = 396, normalized size of antiderivative = 7.20, number of steps used = 12, number of rules used = 6, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6718, 6728, 115, 168, 538, 537} \begin {gather*} -\frac {2 k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 \sqrt {k^2-1} \left (k^2+2 \sqrt {k^2-1}\right ) \sqrt {1-x} \sqrt {x} \sqrt {\frac {k^2 (1-x)}{1-k^2}+1} \Pi \left (\frac {2-k^2}{-k^2-\sqrt {k^2-1}+1};\sin ^{-1}\left (\sqrt {1-x}\right )|-\frac {k^2}{1-k^2}\right )}{\left (2-k^2\right ) \left (-k^2-\sqrt {k^2-1}+1\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {2 \sqrt {k^2-1} \left (k^2-2 \sqrt {k^2-1}\right ) \sqrt {1-x} \sqrt {x} \sqrt {\frac {k^2 (1-x)}{1-k^2}+1} \Pi \left (\frac {2-k^2}{-k^2+\sqrt {k^2-1}+1};\sin ^{-1}\left (\sqrt {1-x}\right )|-\frac {k^2}{1-k^2}\right )}{\left (2-k^2\right ) \left (-k^2+\sqrt {k^2-1}+1\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 115
Rule 168
Rule 537
Rule 538
Rule 6718
Rule 6728
Rubi steps
\begin {align*} \int \frac {1-2 x+k^2 x^2}{\sqrt {(1-x) x \left (1-k^2 x\right )} \left (-1+2 x+\left (-2+k^2\right ) x^2\right )} \, dx &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1-2 x+k^2 x^2}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (-1+2 x+\left (-2+k^2\right ) x^2\right )} \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=\frac {\left (\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \left (-\frac {k^2}{\left (2-k^2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}}+\frac {2 \left (-1+k^2\right ) (1-2 x)}{\left (-2+k^2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (-1+2 x+\left (-2+k^2\right ) x^2\right )}\right ) \, dx}{\sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=-\frac {\left (k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}} \, dx}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \left (1-k^2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1-2 x}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (-1+2 x+\left (-2+k^2\right ) x^2\right )} \, dx}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=-\frac {2 k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \left (1-k^2\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \left (\frac {-2+\frac {k^2}{\sqrt {-1+k^2}}}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (2-2 \sqrt {-1+k^2}+2 \left (-2+k^2\right ) x\right )}+\frac {-2-\frac {k^2}{\sqrt {-1+k^2}}}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (2+2 \sqrt {-1+k^2}+2 \left (-2+k^2\right ) x\right )}\right ) \, dx}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=-\frac {2 k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \left (1-k^2\right ) \left (-2-\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (2+2 \sqrt {-1+k^2}+2 \left (-2+k^2\right ) x\right )} \, dx}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {\left (2 \left (1-k^2\right ) \left (-2+\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} \left (2-2 \sqrt {-1+k^2}+2 \left (-2+k^2\right ) x\right )} \, dx}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=-\frac {2 k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (4 \left (1-k^2\right ) \left (-2-\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1-k^2+k^2 x^2} \left (-2 \left (1-k^2-\sqrt {-1+k^2}\right )+2 \left (2-k^2\right ) x^2\right )} \, dx,x,\sqrt {1-x}\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (4 \left (1-k^2\right ) \left (-2+\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1-k^2+k^2 x^2} \left (-2 \left (1-k^2+\sqrt {-1+k^2}\right )+2 \left (2-k^2\right ) x^2\right )} \, dx,x,\sqrt {1-x}\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=-\frac {2 k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (4 \left (1-k^2\right ) \left (-2-\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {1+\frac {k^2 (-1+x)}{-1+k^2}} \sqrt {1-x} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {k^2 x^2}{1-k^2}} \left (-2 \left (1-k^2-\sqrt {-1+k^2}\right )+2 \left (2-k^2\right ) x^2\right )} \, dx,x,\sqrt {1-x}\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {\left (4 \left (1-k^2\right ) \left (-2+\frac {k^2}{\sqrt {-1+k^2}}\right ) \sqrt {1+\frac {k^2 (-1+x)}{-1+k^2}} \sqrt {1-x} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {k^2 x^2}{1-k^2}} \left (-2 \left (1-k^2+\sqrt {-1+k^2}\right )+2 \left (2-k^2\right ) x^2\right )} \, dx,x,\sqrt {1-x}\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ &=-\frac {2 k^2 \sqrt {1-x} \sqrt {x} \sqrt {1-k^2 x} F\left (\sin ^{-1}\left (\sqrt {x}\right )|k^2\right )}{\left (2-k^2\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}+\frac {2 \sqrt {-1+k^2} \left (k^2+2 \sqrt {-1+k^2}\right ) \sqrt {1+\frac {k^2 (1-x)}{1-k^2}} \sqrt {1-x} \sqrt {x} \Pi \left (\frac {2-k^2}{1-k^2-\sqrt {-1+k^2}};\sin ^{-1}\left (\sqrt {1-x}\right )|-\frac {k^2}{1-k^2}\right )}{\left (2-k^2\right ) \left (1-k^2-\sqrt {-1+k^2}\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}-\frac {2 \left (2-k^2 \left (2-\sqrt {-1+k^2}\right )\right ) \sqrt {1+\frac {k^2 (1-x)}{1-k^2}} \sqrt {1-x} \sqrt {x} \Pi \left (\frac {2-k^2}{1-k^2+\sqrt {-1+k^2}};\sin ^{-1}\left (\sqrt {1-x}\right )|-\frac {k^2}{1-k^2}\right )}{\left (2-k^2\right ) \left (1-k^2+\sqrt {-1+k^2}\right ) \sqrt {(1-x) x \left (1-k^2 x\right )}}\\ \end {align*}
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Mathematica [C] time = 2.92, size = 202, normalized size = 3.67 \begin {gather*} \frac {2 i \sqrt {\frac {1}{x-1}+1} (x-1)^{3/2} \sqrt {\frac {1-\frac {1}{k^2}}{x-1}+1} \left (\left (k^2-2\right ) F\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )+\left (\sqrt {k^2-1}+1\right ) \Pi \left (\frac {k^2-1}{k^2-\sqrt {k^2-1}-1};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )-\left (\sqrt {k^2-1}-1\right ) \Pi \left (\frac {k^2-1}{k^2+\sqrt {k^2-1}-1};i \sinh ^{-1}\left (\frac {1}{\sqrt {x-1}}\right )|1-\frac {1}{k^2}\right )\right )}{\left (k^2-2\right ) \sqrt {(x-1) x \left (k^2 x-1\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 55, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {-2+k^2} \sqrt {x+\left (-1-k^2\right ) x^2+k^2 x^3}}{-1+k^2 x}\right )}{\sqrt {-2+k^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 269, normalized size = 4.89 \begin {gather*} \left [-\frac {\sqrt {-k^{2} + 2} \log \left (\frac {{\left (k^{4} - 4 \, k^{2} + 4\right )} x^{4} - 4 \, {\left (2 \, k^{4} - 5 \, k^{2} + 2\right )} x^{3} + 2 \, {\left (4 \, k^{4} - 5 \, k^{2} - 4\right )} x^{2} - 4 \, \sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left ({\left (k^{2} - 2\right )} x^{2} - 2 \, {\left (k^{2} - 1\right )} x + 1\right )} \sqrt {-k^{2} + 2} - 4 \, {\left (2 \, k^{2} - 3\right )} x + 1}{{\left (k^{4} - 4 \, k^{2} + 4\right )} x^{4} + 4 \, {\left (k^{2} - 2\right )} x^{3} - 2 \, {\left (k^{2} - 4\right )} x^{2} - 4 \, x + 1}\right )}{2 \, {\left (k^{2} - 2\right )}}, \frac {\arctan \left (\frac {\sqrt {k^{2} x^{3} - {\left (k^{2} + 1\right )} x^{2} + x} {\left ({\left (k^{2} - 2\right )} x^{2} - 2 \, {\left (k^{2} - 1\right )} x + 1\right )} \sqrt {k^{2} - 2}}{2 \, {\left ({\left (k^{4} - 2 \, k^{2}\right )} x^{3} - {\left (k^{4} - k^{2} - 2\right )} x^{2} + {\left (k^{2} - 2\right )} x\right )}}\right )}{\sqrt {k^{2} - 2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {k^{2} x^{2} - 2 \, x + 1}{\sqrt {{\left (k^{2} x - 1\right )} {\left (x - 1\right )} x} {\left ({\left (k^{2} - 2\right )} x^{2} + 2 \, x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.25, size = 2705, normalized size = 49.18
method | result | size |
default | \(-\frac {2 \sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}\, \sqrt {\frac {-1+x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticF \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (k^{2}-2\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}+\frac {-\frac {8 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {-1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (-\frac {2 k^{2}}{k^{2}-2}+\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}-\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}-\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )}+\frac {8 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {-1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (-\frac {2 k^{2}}{k^{2}-2}+\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}-\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}-\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )}+\frac {8 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {-1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) \sqrt {k^{2}-1}}{\left (-\frac {2 k^{2}}{k^{2}-2}+\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}-\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}-\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )}-\frac {8 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {-1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) \sqrt {k^{2}-1}}{\left (-\frac {2 k^{2}}{k^{2}-2}+\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}-\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}-\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )}-\frac {4 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {-1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (-\frac {2 k^{2}}{k^{2}-2}+\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}-\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}-\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right )}+\frac {4 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {-1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (-\frac {2 k^{2}}{k^{2}-2}+\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}-\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}-\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right )}-\frac {8 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (-\frac {2 k^{2}}{k^{2}-2}-\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}+\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}+\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )}+\frac {8 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (-\frac {2 k^{2}}{k^{2}-2}-\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}+\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}+\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )}-\frac {8 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) \sqrt {k^{2}-1}}{\left (-\frac {2 k^{2}}{k^{2}-2}-\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}+\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}+\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )}+\frac {8 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) \sqrt {k^{2}-1}}{\left (-\frac {2 k^{2}}{k^{2}-2}-\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}+\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}+\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )}-\frac {4 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (-\frac {2 k^{2}}{k^{2}-2}-\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}+\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}+\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right )}+\frac {4 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (-\frac {2 k^{2}}{k^{2}-2}-\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}+\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}+\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right )}}{k^{2}-2}\) | \(2705\) |
elliptic | \(-\frac {2 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticF \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (k^{2}-2\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}}-\frac {8 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {-1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (-\frac {2 k^{2}}{k^{2}-2}+\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}-\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}-\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )^{2}}+\frac {8 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {-1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (-\frac {2 k^{2}}{k^{2}-2}+\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}-\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}-\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )^{2}}+\frac {8 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {-1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) \sqrt {k^{2}-1}}{\left (-\frac {2 k^{2}}{k^{2}-2}+\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}-\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}-\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )^{2}}-\frac {8 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {-1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) \sqrt {k^{2}-1}}{\left (-\frac {2 k^{2}}{k^{2}-2}+\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}-\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}-\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )^{2}}-\frac {4 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {-1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (-\frac {2 k^{2}}{k^{2}-2}+\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}-\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}-\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )}+\frac {4 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}-\frac {-1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (-\frac {2 k^{2}}{k^{2}-2}+\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}-\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}-\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )}-\frac {8 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (-\frac {2 k^{2}}{k^{2}-2}-\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}+\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}+\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )^{2}}+\frac {8 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (-\frac {2 k^{2}}{k^{2}-2}-\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}+\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}+\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )^{2}}-\frac {8 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) \sqrt {k^{2}-1}}{\left (-\frac {2 k^{2}}{k^{2}-2}-\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}+\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}+\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )^{2}}+\frac {8 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right ) \sqrt {k^{2}-1}}{\left (-\frac {2 k^{2}}{k^{2}-2}-\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}+\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}+\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )^{2}}-\frac {4 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (-\frac {2 k^{2}}{k^{2}-2}-\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}+\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}+\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )}+\frac {4 \sqrt {-k^{2} x +1}\, \sqrt {-\frac {1}{\frac {1}{k^{2}}-1}+\frac {x}{\frac {1}{k^{2}}-1}}\, \sqrt {k^{2} x}\, \EllipticPi \left (\sqrt {-\left (x -\frac {1}{k^{2}}\right ) k^{2}}, \frac {1}{k^{2} \left (\frac {1}{k^{2}}+\frac {1+\sqrt {k^{2}-1}}{k^{2}-2}\right )}, \sqrt {\frac {1}{k^{2} \left (\frac {1}{k^{2}}-1\right )}}\right )}{\left (-\frac {2 k^{2}}{k^{2}-2}-\frac {2 k^{2} \sqrt {k^{2}-1}}{k^{2}-2}+\frac {4}{k^{2}-2}+\frac {4 \sqrt {k^{2}-1}}{k^{2}-2}+2\right ) k^{2} \sqrt {k^{2} x^{3}-k^{2} x^{2}-x^{2}+x}\, \left (\frac {1}{k^{2}}+\frac {1}{k^{2}-2}+\frac {\sqrt {k^{2}-1}}{k^{2}-2}\right ) \left (k^{2}-2\right )}\) | \(2727\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.10, size = 84, normalized size = 1.53 \begin {gather*} \frac {\ln \left (\frac {x\,2{}\mathrm {i}+k^2\,x^2\,1{}\mathrm {i}-k^2\,x\,2{}\mathrm {i}-x^2\,2{}\mathrm {i}-2\,\sqrt {k^2-2}\,\sqrt {x\,\left (k^2\,x-1\right )\,\left (x-1\right )}+1{}\mathrm {i}}{2\,k^2\,x^2-4\,x^2+4\,x-2}\right )\,1{}\mathrm {i}}{\sqrt {k^2-2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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